1.
DESIGN AND SIMULATION OF
PHOTOVOLTAIC WATER PUMPING SYSTEM
A Thesis
Presented to the Faculty of
California Polytechnic State University,
San Luis Obispo
In Partial Fulfillment
of the Requirements for the Degree of
Master of Science in Electrical Engineering
by
Akihiro Oi
September 2005
2.
AUTHORIZATION FOR REPRODUCTION OF MASTER’S THESIS
I grant permission for the reproduction of this thesis in its entirety or any of its parts, without
further authorization from me.
Signature (Akihiro Oi)
Date
ii
3.
APPROVAL
Title: DESIGN AND SIMULATION OF PHOTOVOLTAIC WATER PUMPING SYSTEM
Author:
Akihiro Oi
Date Submitted:
26th September, 2005
Dr. Taufik
Committee Chair
Signature
Dr. Ahmad Nafisi
Committee Member
Signature
Dr. William Ahlgren
Committee Member
Signature
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4.
ABSTRACT
DESIGN AND SIMULATION OF PHOTOVOLTAIC WATER PUMPING SYSTEM
Akihiro Oi
This thesis deals with the design and simulation of a simple but efficient photovoltaic
water pumping system.
It provides theoretical studies of photovoltaics and modeling
techniques using equivalent electric circuits. The system employs the maximum power point
tracker (MPPT). The investigation includes discussion of various MPPT algorithms and
control methods.
PSpice simulations verify the DCDC converter design.
MATLAB
simulations perform comparative tests of two popular MPPT algorithms using actual
irradiance data. The thesis decides on the output sensing direct control method because it
requires fewer sensors. This allows a lower cost system. Each subsystem is modeled in
order to simulate the whole system in MATLAB. It employs SIMULINK to model a DC
pump motor, and the model is transferred into MATLAB. Then, MATLAB simulations
verify the system and functionality of MPPT. Simulations also make comparisons with the
system without MPPT in terms of total energy produced and total volume of water pumped
per day. The results validate that MPPT can significantly increase the efficiency and the
performance of PV water pumping system compared to the system without MPPT.
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5.
ACKNOWLEDGEMENTS
I would like to first acknowledge my advisor, Dr. Taufik, for his support and advice
throughout my graduate program. His power electronics courses and his dedication to his
students gave me the best experience during the program. I would also like to express my
sincere appreciation to my other thesis committees, Dr. Nafisi and Dr. Ahlgren, for review of
this thesis in detail and their important feedback.
I would like to thank my colleague and friend, John Carlin, who has a career
experience in designing photovoltaic systems.
A number of ideas generated from our
numerous discussions and his feedback are incorporated in this thesis. Also, thanks to my
other good colleagues, James Silva, John Cadwell, Michael Chong, James Sorenson, Sajiv
Nair, Alan Yeung, Yat Tam, all other denizens of “EE Grad Lab” and the lab technicians for
their support and willingness to help me out during various stages of my project.
Finally, to my parents, my sister, and my friends  many thanks for much support the
whole way through, especially Jenny Ho for her constant encouragement and support during
two years of my graduate work.
Akihiro Oi
September 2005
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6.
Table of Contents
List of Tables .................................................................................................... viii
List of Figures ......................................................................................................ix
Chapter 1 Introduction ..........................................................................................1
1.1 Water Pumping Systems and Photovoltaic Power.......................................................... 1
1.2 Energy Storage Alternatives ........................................................................................... 3
1.3 The Proposed System...................................................................................................... 4
1.3.1 PV Module............................................................................................................................ 5
1.3.2 Maximum Power Point Tracker............................................................................................ 5
1.3.3 MPPT Controller .................................................................................................................. 6
1.3.4 Water Pump .......................................................................................................................... 7
1.4 Background and Scope of This Thesis............................................................................ 8
Chapter 2 Photovoltaic Modules ........................................................................10
2.1 Introduction................................................................................................................... 10
2.2 Photovoltaic Cell........................................................................................................... 10
2.3 Modeling a PV Cell ...................................................................................................... 12
2.3.1 The Simplest Model............................................................................................................ 12
2.3.2 The More Accurate Model.................................................................................................. 15
2.4 Photovoltaic Module..................................................................................................... 17
2.5 Modeling a PV Module by MATLAB.......................................................................... 18
2.6 The IV Curve and Maximum Power Point.................................................................. 25
Chapter 3 Maximum Power Point Tracker.........................................................27
3.1 Introduction................................................................................................................... 27
3.2 IV Characteristics of DC Motors................................................................................. 28
3.3 DCDC Converter ......................................................................................................... 31
3.3.1 Topologies .......................................................................................................................... 31
3.3.2 Cúk and SEPIC Converters ................................................................................................ 32
3.3.3 Basic Operation of Cúk Converter...................................................................................... 34
3.4 Mechanism of Load Matching ...................................................................................... 37
3.5 Maximum Power Point Tracking Algorithms............................................................... 38
3.5.1 Perturb & Observe Algorithm............................................................................................. 40
3.5.2 Incremental Conductance Algorithm.................................................................................. 44
3.6 Control of MPPT........................................................................................................... 47
3.6.1 PI Control............................................................................................................................ 47
3.6.2 Direct Control ..................................................................................................................... 48
3.6.3 Output Sensing Direct Control ........................................................................................... 50
3.7 Limitations of MPPT .................................................................................................... 52
Chapter 4 Design and Simulations .....................................................................55
4.1 Introduction................................................................................................................... 55
4.2 Cúk Converter Design................................................................................................... 55
4.2.1 Component Selection.......................................................................................................... 56
4.2.2 PSpice Simulations ............................................................................................................. 59
4.2.3 Choice of MPPT Sampling Rate......................................................................................... 61
4.3 Comparisons of P&O and incCond Algorithm............................................................. 62
4.4 MPPT Simulations with Resistive Load ....................................................................... 66
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4.5 MPPT Simulations with DC Pump Motor Load........................................................... 70
4.5.1 Modeling of DC Water Pump............................................................................................. 71
4.5.2 MATLAB Simulation Results ............................................................................................ 73
4.6 System with MPPT vs. Directcoupled System............................................................ 75
Chapter 5 Conclusion..........................................................................................78
5.1 Summary ....................................................................................................................... 78
5.2 Difficulties and Future Research .................................................................................. 79
5.3 Concluding Remarks..................................................................................................... 80
Bibliography .......................................................................................................81
Appendix A.........................................................................................................84
A.1 MATLAB Functions and Scripts ................................................................................. 84
A.1.1 MATLAB Function for Modeling BP SX 150S PV Module............................................. 84
A.1.2 MATLAB Script to Draw PV IV Curves ......................................................................... 85
A.1.3 MATLAB Function to Find the MPP ................................................................................ 86
A.1.4 MATLAB Script: P&O Algorithm .................................................................................... 86
A.1.5 MATLAB Script: incCond Algorithm............................................................................... 88
A.1.6 MATLAB Script for MPPT with Output Sensing Direct Control Method........................ 90
A.1.7 MATLAB Script for MPPT Simulations with DC Pump Motor Load.............................. 93
A.1.8 MATLAB Script for MPPT Simulations with Directcoupled DC Water Pump .............. 97
A.2 MPPT Simulations with Resistive Load .................................................................... 100
A.2.1 Direct Control Method with P&O Algorithm.................................................................. 100
A.2.2 Direct Control Method with incCond Algorithm............................................................. 101
Appendix B .......................................................................................................102
B.1 DSP Control ............................................................................................................... 102
B.1.1 TMS320F2812 DSP......................................................................................................... 102
B.1.2 SIMULNK and TI DSP.................................................................................................... 102
B.1.3 Example ........................................................................................................................... 103
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8.
List of Tables
Table 11: PV powered, Diesel powered, vs. Windmill [13].................................................... 3
Table 21: Electrical characteristics data of PV module taken from the datasheet [1]........... 18
Table 31: Load matching with the resistive load (6 ) under the varying irradiance............ 53
Table 32: Load matching with the resistive load (12 ) under the varying irradiance.......... 53
Table 41: Design specification of the Cúk Converter ........................................................... 55
Table 42: Cúk converter design: comparisons of simulations and calculated results ........... 60
Table 43: Comparison of the P&O and incCond algorithms on a cloudy day ...................... 65
Table 44: Energy production and efficiency of PV module with and without MPPT .......... 75
Table 45: Total volume of water pumped for 12 hours......................................................... 77
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9.
List of Figures
Figure 11: Block diagram of the proposed PV water pumping system................................... 5
Figure 21: Illustration of the pn junction of PV cell [16]..................................................... 11
Figure 22: Illustrated side view of solar cell and the conducting current [16]...................... 11
Figure 23: PV cell with a load and its simple equivalent circuit [16] ................................... 12
Figure 24: Diagrams showing a shortcircuit and an opencircuit condition [16]................. 13
Figure 25: IV plot of ideal PV cell under two different levels of irradiance (25oC)............ 15
Figure 26: More accurate equivalent circuit of PV cell......................................................... 16
Figure 27: PV cells are connected in series to make up a PV module .................................. 17
Figure 28: Picture of BP SX 150S PV module [1] ................................................................ 18
Figure 29: Equivalent circuit used in the MATLAB simulations ......................................... 19
Figure 210: Effect of diode ideally factors by MATLAB simulation (1KW/m2, 25oC) ....... 21
Figure 211: Effect of series resistances by MATLAB simulation (1KW/m2, 25oC) ............ 22
Figure 212: IV curves of BP SX 150S PV module at various temperatures........................ 24
Figure 213: Simulated IV curve of BP SX 150S PV module (1KW/m2, 25oC) .................. 25
Figure 214: IV and PV relationships of BP SX 150S PV module...................................... 26
Figure 31: PV module is directly connected to a (variable) resistive load............................ 27
Figure 32: IV curves of BP SX 150S PV module and various resistive loads..................... 28
Figure 33: Electrical model of permanent magnet DC motor ............................................... 29
Figure 34: PV IV curves with varying irradiance and a DC motor IV curve ..................... 30
Figure 35: PV IV curves with isopower lines (dotted) and a DC motor IV curve ............ 31
Figure 36: Circuit diagram of the basic Cúk converter ......................................................... 34
Figure 37: Circuit diagram of the basic SEPIC converter ..................................................... 34
Figure 38: Basic Cúk converter when the switch is ON........................................................ 35
Figure 39: Basic Cúk converter when the switch is OFF ...................................................... 35
Figure 310: The impedance seen by PV is Rin that is adjustable by duty cycle (D).............. 38
Figure 311: IV curves for varying irradiance and a trace of MPPs (25oC).......................... 39
Figure 312: IV curves for varying irradiance and a trace of MPPs (50oC).......................... 40
Figure 313: Plot of power vs. voltage for BP SX 150S PV module (1KW/m2, 25oC).......... 41
Figure 314: Flowchart of the P&O algorithm ....................................................................... 41
Figure 315: Erratic behavior of the P&O algorithm under rapidly increasing irradiance..... 43
Figure 316: Flowchart of the incCond algorithm .................................................................. 46
Figure 317: Block diagram of MPPT with the PI compensator ............................................ 48
Figure 318: Block diagram of MPPT with the direct control................................................ 48
Figure 319: Relationship of the input impedance of Cúk converter and its duty cycle ........ 49
Figure 320: Output power of Cúk converter vs. its duty cycle (1KW/m2, 25oC).................. 51
Figure 321: Flowchart of P&O algorithm for the output sensing direct control method ...... 52
Figure 41: Schematic of the Cúk converter with PMDC motor load .................................... 59
Figure 42: PSpice plots of input/output current (above) and voltage (below) ...................... 60
Figure 43: Transient response when duty cycle is increased 0.35% at 250ms...................... 61
Figure 44: Searching the MPP (1KW/m2, 25oC)................................................................... 62
Figure 45: Irradiance data for a sunny and a cloudy day of April in Barcelona, Spain [2]... 63
Figure 46: Traces of MPP tracking on a sunny day (25oC)................................................... 64
Figure 47: Trace of MPP tracking on a cloudy day (25oC) ................................................... 65
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10.
Figure 48: MPPT simulation flowchart ................................................................................. 68
Figure 49: MPPT simulations with the resistive load (100 to 1000W/m2, 25oC) ................. 69
Figure 410: Output protection & regulation (100 to 1000W/m2, 25oC)................................ 70
Figure 411: Kyocera SD 1230 water pump performance chart [13].................................... 71
Figure 412: SIMULINK model of permanent magnet DC pump motor............................... 72
Figure 413: SIMULINK DC machine block parameters....................................................... 72
Figure 414: SIMULINK plot of Rload ( ).............................................................................. 73
Figure 415: MPPT simulations with the DC pump motor load (20 to 1000W/m2, 25oC)..... 74
Figure 416: SIMULINK plot of DC motor IV curve ........................................................... 75
Figure 417: Flow rates of PV water pumps for a 12hour period.......................................... 76
Figure A1: MPPT Simulations with the direct control method (P&O algorithm) .............. 100
Figure A2: MPPT Simulations with the direct control method (incCond algorithm) ......... 101
Figure B1: A simple example of generating PWM from the voltage input ........................ 103
Figure B2: Plots of the input voltage and the PWM output shown as duty cycle ............... 103
x
11.
Chapter 1 Introduction
Water resources are essential for satisfying human needs, protecting health, and
ensuring food production, energy and the restoration of ecosystems, as well as for social and
economic development and for sustainable development [25]. However, according to UN
World Water Development Report in 2003, it has been estimated that two billion people are
affected by water shortages in over forty countries, and 1.1 billion do not have sufficient
drinking water [26]. There is a great and urgent need to supply environmentally sound
technology for the provision of drinking water. Remote water pumping systems are a key
component in meeting this need. It will also be the first stage of the purification and
desalination plants to produce potable water.
In this thesis, a simple but efficient photovoltaic water pumping system is presented.
It provides theoretical studies of photovoltaics (PV) and its modeling techniques. It also
investigates in detail the maximum power point tracker (MPPT), a power electronic device
that significantly increases the system efficiency. At last, it presents MATLAB simulations
of the system and makes comparisons with a system without MPPT.
1.1 Water Pumping Systems and Photovoltaic Power
A water pumping system needs a source of power to operate.
In general, AC
powered system is economic and takes minimum maintenance when AC power is available
from the nearby power grid. However, in many rural areas, water sources are spread over
many miles of land and power lines are scarce. Installation of a new transmission line and a
transformer to the location is often prohibitively expensive. Windmills have been installed
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12.
traditionally in such areas; many of them are, however, inoperative now due to lack of proper
maintenance and age. Today, many standalone type water pumping systems use internal
combustion engines. These systems are portable and easy to install. However, they have
some major disadvantages, such as: they require frequent site visits for refueling and
maintenance, and furthermore diesel fuel is often expensive and not readily available in rural
areas of many developing countries.
The consumption of fossil fuels also has an environmental impact, in particular the
release of carbon dioxide (CO2) into the atmosphere. CO2 emissions can be greatly reduced
through the application of renewable energy technologies, which are already cost competitive
with fossil fuels in many situations. Good examples include largescale gridconnected wind
turbines, solar water heating, and offgrid standalone PV systems [24].
The use of
renewable energy for water pumping systems is, therefore, a very attractive proposition.
Windmills are a longestablished method of using renewable energy; however they
are quickly phasing out from the scene despite success of largescale gridtied wind turbines.
PV systems are highly reliable and are often chosen because they offer the lowest
lifecycle cost, especially for applications requiring less than 10KW, where grid electricity is
not available and where internalcombustion engines are expensive to operate [24]. If the
water source is 1/3 mile (app. 0.53Km) or more from the power line, PV is a favorable
economic choice [13]. Table 11 shows the comparisons of different standalone type water
pumping systems.
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13.
System Type
PV Powered System
Diesel (or Gas) Powered
System
Windmill
Advantages
Low maintenance
Unattended operation
Reliable long life
No fuel and no fumes
Easy to install
Low recurrent costs
System is modular and
closely matched to need
Moderate capital costs
Easy to install
Can be portable
Extensive experience
available
No fuel and no fumes
Potentially longlasting
Works well in windy sites
Disadvantages
Relatively high initial cost
Low output in cloudy
weather
Needs maintenance and
replacement
Site visits necessary
Noise, fume, dirt problems
Fuel often expensive and
supply intermittent
High maintenance
Seasonal disadvantages
Difficult find parts thus
costly repair
Installation is labor
intensive and needs special
tools
Table 11: PV powered, Diesel powered, vs. Windmill [13]
1.2 Energy Storage Alternatives
Needless to say, photovoltaics are able to produce electricity only when the sunlight
is available, therefore standalone systems obviously need some sort of backup energy
storage which makes them available through the night or bad weather conditions.
Among many possible storage technologies, the leadacid battery continues to be the
workhorse of many PV systems because it is relatively inexpensive and widely available. In
addition to energy storage, the battery also has ability to provide surges of current that are
much higher than the instantaneous current available from the array, as well as the inherent
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and automatic property controlling the output voltage of the array so that loads receive
voltages within their own range of acceptability [16].
While batteries may seem like a good idea, they have a number of disadvantages.
The type of leadacid battery suitable for PV systems is a deepcycle battery [15], which is
different from one used for automobiles, and it is more expensive and not widely available.
Battery lifetime in PV systems is typically three to eight years, but this reduces to typically
two to six years in hot climate since high ambient temperature dramatically increases the rate
of internal corrosion [24]. Batteries also require regular maintenance and will degrade very
rapidly if the electrolyte is not topped up and the charge is not maintained. They reduce the
efficiency of the overall system due to power loss during charge and discharge. Typical
battery efficiency is around 85% but could go below 75% in hot climate [24]. From all those
reasons, experienced PV system designers avoid batteries whenever possible.
For water pumping systems, appropriately sized water reservoirs can meet the
requirement of energy storage during the downtime of PV generation. The additional cost of
reservoir is considerably lower than that incurred by the battery equipped system. As a
matter of fact, only about five percent of solar pumping systems employ a battery bank [13].
1.3 The Proposed System
The experimental water pumping system proposed in this thesis is a standalone type
without backup batteries. As shown in Figure 11, the system is very simple and consists of
a single PV module, a maximum power point tracker (MPPT), and a DC water pump. The
size of the system is intended to be small; therefore it could be built in the lab in the future.
The system including the subsystems will be simulated to verify the functionalities.
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15.
DC Water
Pump [13]
PV Module [1]
Figure 11: Block diagram of the proposed PV water pumping system
1.3.1 PV Module
There are different sizes of PV module commercially available (typically sized from
60W to 170W). Usually, a number of PV modules are combined as an array to meet
different energy demands. For example, a typical smallscale desalination plant requires a
few thousand watts of power [24]. The size of system selected for the proposed system is
150W, which is commonly used in small water pumping systems for cattle grazing in rural
areas of the United States. The power electronics lab located in the building 20, room 104,
has three BP SX 150S multicrystalline PV modules. Each module provides a maximum
power of 150W [13], therefore the proposed system requires only one of them. A detailed
discussion about PV and modeling of PV appears in Chapter 2.
1.3.2 Maximum Power Point Tracker
The maximum power point tracker (MPPT) is now prevalent in gridtied PV power
systems and is becoming more popular in standalone systems. It should not be confused
with sun trackers, mechanical devices that rotate and/or tilt PV modules in the direction of
sun. MPPT is a power electronic device interconnecting a PV power source and a load,
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16.
maximizes the power output from a PV module or array with varying operating conditions,
and therefore maximizes the system efficiency. MPPT is made up with a switchmode DCDC converter and a controller. For gridtied systems, a switchmode inverter sometimes fills
the role of MPPT. Otherwise, it is combined with a DCDC converter that performs the
MPPT function.
In addition to MPPT, the system could also employ a sun tracker. According to the
data in reference [15], the singleaxis sun tracker can collect about 40% more energy than a
seasonally optimized fixedaxis collector in summer in a dry climate such as Albuquerque,
New Mexico. In winter, however, it can gain only 20% more energy. In a climate with more
water vapor in the atmosphere such as Seattle, Washington, the effect of sun tracker is
smaller because a larger fraction of solar irradiation is diffuse. It collects 30% more energy
in summer, but the gain is less than 10% in winter. The twoaxis tracker is only a few
percent better than the singleaxis version. Sun tracking enables the system to meet energy
demand with smaller PV modules, but it increases the cost and complexity of system. Since
it is made of moving parts, there is also a higher chance of failure. Therefore, in this simple
system, the sun tracker is not implemented. A detailed discussion on MPPT appears in
Chapter 3.
1.3.3 MPPT Controller
Analog controllers have traditionally performed control of MPPT. However, the use
of digital controllers is rapidly increasing because they offer several advantages over analog
controllers.
First, digital controllers are programmable thus capable of implementing
advanced algorithm with relative ease. It is far easier to code the equation, x = y × z, than to
design an analog circuit to do the same [23]. For the same reason, modification of the design
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is much easier with digital controllers. They are immune to time and temperature drifts
because they work in discrete, outside the linear operation. As a result, they offer longterm
stability. They are also insensitive to component tolerances since they implement algorithm
in software, where gains and parameters are consistent and reproducible [23]. They allow
reduction of parts count since they can handle various tasks in a single chip. Many of them
are also equipped with multiple A/D converters and PWM generators, thus they can control
multiple devices with a single controller.
This thesis, therefore, chooses a method of digital control for MPPT. The design and
simulations of MPPT in Chapter 4 are done on the premise that it is going to be built with a
microcontroller or a DSP, and the algorithm is readily transferable to its implementation.
Chapter 3 provides discussions of various control methods.
Appendix B provides
introduction of Texas Instruments DSP and SIMULINK as an implementation tool.
1.3.4 Water Pump
Two types of pumps are commonly used for PV water pumping applications: positive
displacement and centrifugal [19]. Positive displacement types are used in lowvolume
pumps [13] and costeffective. Centrifugal pumps have relatively high efficiency [19] and
are capable of pumping a high volume of water [13]. A typical size of system with this type
pump is at least 500W or larger. There is a growing trend among the pump manufacturers to
use them with brushless DC motors (BDCM) for higher efficiency and low maintenance [19].
However, the cost and complexity of these systems will be significantly higher. Water
pumps are driven by various types of motors. AC induction motors are cheaper and widely
available worldwide. The system, however, needs an inverter to convert DC output power
from PV to AC power, which is usually expensive, and it is also less efficient than DC
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18.
motorpump systems [19]. In general, DC motors are preferred because they are highly
efficient and can be directly coupled with a PV module or array. Brushed types are less
expensive and more common although brushes need to be replaced periodically (typically
every two years) [19]. There is also an aforementioned brushless type.
The water pump chosen here for its size and cost is the Kyocera SD 1230
submersible solar pump, pictured in Figure 11. It is a diaphragmtype positive displacement
pump equipped with a brushed permanent magnet DC motor and designed for use in standalone water delivery systems, specifically for water delivery in remote locations. Flow rates
up to 17.0L/min (4.5GPM) and heads up to 30.0m (100ft.) [13]. The typical daily output is
between 2,700L and 5,000L [13]. The rated maximum power consumption is 150W. It
operates with a low voltage (12~30V DC), and its power requirement is as little as 35W [13].
A simple model of this water pump is used for simulations in Chapter 4.
1.4 Background and Scope of This Thesis
The impetus for this research is to investigate the use of power electronics in
renewable energy, particularly photovoltaics (PV). Numerous studies have been done in PV
systems, a significant number of them in Europe, Japan, and Australia. In the United Sates,
there is a growing interest in PV, but research and development in PV systems is far behind
from the aforementioned countries, and unfortunately, California State Universities (CSUs)
are no exception. There are only a small number of studies related to PV systems in the past.
Among them, there were a few senior projects which built PV facilities here in California
Polytechnic State University, San Luis Obispo. Two senior projects, also here, built a simple
PV battery charger, and a few others dealt with a sun tracker. There have been only two
master’s theses written about PV systems in the CSU system. The first attempt to study
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19.
MPPT was made by Dang [4] of California Polytechnic State University, Pomona. The
thesis built a small PV module simulator and a buck converter without a controller. Then, it
provided a rudimentary computer simulation of MPPT with a resistive load. The study was,
however, far from comprehensive. Another was done here by Day [5], and it centered round
a power system for a miniature satellite.
It included MPPT in the system, but the
functionality of MPPT was not tested. The theoretical study was insufficient, and it lacked
simulations and experiments to ensure the functionality of MPPT.
MPPT is one of many applications of power electronics, and it is a relatively new and
unknown area. There is no textbook that provides comprehensive and detailed explanations
about MPPT. Therefore, this thesis investigates it in detail and provides better explanations
for students who are interested in this research area. In order to understand and design
MPPT, it is necessary to have a good understanding of the behaviors of PV. The thesis
facilitates it using MATLAB models of PV cell and module. Each subsystem in the PV
water pumping system is modeled for MATLAB simulations. Finally, the functionality of
MPPT for water pumping systems is verified and validated.
This thesis is limited to providing theoretical studies and simulations of PV water
pumping system with MPPT. The system will not be built in this thesis; that is left as future
work.
Thus, it will not cover a discussion about actual implementation of DSP or
microcontrollers, nor other hardware implementation, beyond a discussion on component
selection for the DCDC converter. A major assumption made in simulations is the use of an
ideal DCDC converter, as opposed to a more realistic model that includes losses. The model,
however, should provide sufficient results for verification of MPPT functionality.
9
20.
Chapter 2 Photovoltaic Modules
2.1 Introduction
The history of PV dates back to 1839 when a French physicist, Edmund Becquerel,
discovered the first photovoltaic effect when he illuminated a metal electrode in an
electrolytic solution [16]. Thirtyseven years later British physicist, William Adams, with his
student, Richard Day, discovered a photovoltaic material, selenium, and made solid cells
with 1~2% efficiency which were soon widely adopted in the exposure meters of camera [16].
In 1954 the first generation of semiconductor siliconbased PV cells was born, with
efficiency of 6% [3], and adopted in space applications. Today, the production of PV cells is
following an exponential growth curve since technological advancement of late ‘80s that has
started to rapidly improve efficiency and reduce cost.
This chapter discusses the fundamentals of PV cells and modeling of a PV cell using
an equivalent electrical circuit. The models are implemented using MATLAB to study PV
characteristics and simulate a real PV module.
2.2 Photovoltaic Cell
Photons of light with energy higher than the bandgap energy of PV material can
make electrons in the material break free from atoms that hold them and create holeelectron
pairs, as shown in Figure 21. These electrons, however, will soon fall back into holes
causing charge carriers to disappear. If a nearby electric field is provided, those in the
conduction band can be continuously swept away from holes toward a metallic contact where
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21.
they will emerge as an electric current. The electric field within the semiconductor itself at
the junction between two regions of crystals of different type, called a pn junction [16].
Figure 21: Illustration of the pn junction of PV cell [16]
Showing holeelectron pairs created by photons
The PV cell has electrical contacts on its top and bottom to capture the electrons, as
shown in Figure 22. When the PV cell delivers power to the load, the electrons flow out of
the nside into the connecting wire, through the load, and back to the pside where they
recombine with holes [16]. Note that conventional current flows in the opposite direction
from electrons.
Figure 22: Illustrated side view of solar cell and the conducting current [16]
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22.
2.3 Modeling a PV Cell
The use of equivalent electric circuits makes it possible to model characteristics of a
PV cell. The method used here is implemented in MATLAB programs for simulations. The
same modeling technique is also applicable for modeling a PV module.
2.3.1 The Simplest Model
The simplest model of a PV cell is shown as an equivalent circuit below that consists
of an ideal current source in parallel with an ideal diode. The current source represents the
current generated by photons (often denoted as Iph or IL), and its output is constant under
constant temperature and constant incident radiation of light.
Figure 23: PV cell with a load and its simple equivalent circuit [16]
There are two key parameters frequently used to characterize a PV cell. Shorting
together the terminals of the cell, as shown in Figure 24 (a), the photon generated current
will follow out of the cell as a shortcircuit current (Isc). Thus, Iph = Isc. As shown in Figure
24 (b), when there is no connection to the PV cell (opencircuit), the photon generated
current is shunted internally by the intrinsic pn junction diode. This gives the open circuit
voltage (Voc). The PV module or cell manufacturers usually provide the values of these
parameters in their datasheets.
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23.
Figure 24: Diagrams showing a shortcircuit and an opencircuit condition [16]
The output current (I) from the PV cell is found by applying the Kirchoff’s current
law (KCL) on the equivalent circuit shown in Figure 23.
I = I sc − I d
(2.1)
where: Isc is the shortcircuit current that is equal to the photon generated current, and Id is
the current shunted through the intrinsic diode.
The diode current Id is given by the Shockley’s diode equation:
I d = I o (e qVd / kT − 1)
(2.2)
where: Io is the reverse saturation current of diode (A),
q is the electron charge (1.602×1019 C),
Vd is the voltage across the diode (V),
k is the Boltzmann’s constant (1.381×1023 J/K),
T is the junction temperature in Kelvin (K).
Replacing Id of the equation (2.1) by the equation (2.2) gives the currentvoltage
relationship of the PV cell.
I = I sc − I o (e qV / kT − 1)
where: V is the voltage across the PV cell, and I is the output current from the cell.
13
(2.3)
24.
The reverse saturation current of diode (Io) is constant under the constant temperature
and found by setting the opencircuit condition as shown in Figure 24 (b). Using the
equation (2.3), let I = 0 (no output current) and solve for Io.
0 = I sc − I o (e qVoc / kT − 1)
(2.4)
I sc = I o (e qVoc / kT − 1)
(2.5)
Io =
I
(e
sc
qVoc / kT
− 1)
(2.6)
To a very good approximation, the photon generated current, which is equal to Isc, is
directly proportional to the irradiance, the intensity of illumination, to PV cell [15]. Thus, if
the value, Isc, is known from the datasheet, under the standard test condition, Go=1000W/m2
at the air mass (AM) = 1.5, then the photon generated current at any other irradiance, G
(W/m2), is given by:
I sc G =
G
I sc Go
Go
(2.7)
Figure 25 shows that current and voltage relationship (often called as an IV curve)
of an ideal PV cell simulated by MATLAB using the simplest equivalent circuit model. The
discussion of MATLAB simulations will appear in Section 2.5. The PV cell output is both
limited by the cell current and the cell voltage, and it can only produce a power with any
combinations of current and voltage on the IV curve. It also shows that the cell current is
proportional to the irradiance.
14
25.
5
Full Sun (1000W/m2)
4.5
4
Cell Current (A)
3.5
3
2.5
Half Sun (500W/m2)
2
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
Cell Voltage (V)
0.5
0.6
0.7
Figure 25: IV plot of ideal PV cell under two different levels of irradiance (25oC)
2.3.2 The More Accurate Model
There are a few things that have not been taken into account in the simple model and
that will affect the performance of a PV cell in practice.
a) Series Resistance
In a practical PV cell, there is a series of resistance in a current path through the
semiconductor material, the metal grid, contacts, and current collecting bus [2]. These
resistive losses are lumped together as a series resister (Rs).
Its effect becomes very
conspicuous in a PV module that consists of many seriesconnected cells, and the value of
resistance is multiplied by the number of cells.
b) Parallel Resistance
This is also called shunt resistance. It is a loss associated with a small leakage of
current through a resistive path in parallel with the intrinsic device [2].
15
This can be
26.
represented by a parallel resister (Rp). Its effect is much less conspicuous in a PV module
compared to the series resistance, and it will only become noticeable when a number of PV
modules are connected in parallel for a larger system.
c) Recombination
Recombination in the depletion region of PV cells provides nonohmic current paths
in parallel with the intrinsic PV cell [2] [7]. As shown in Figure 26, this can be represented
by the second diode (D2) in the equivalent circuit.
Rs
+
Isc
D1
D2
n=1
Rp
n=2
V
Load
Figure 26: More accurate equivalent circuit of PV cell
Summarizing these effects, the currentvoltage relationship of PV cell is written as:
I = I sc − I o1 e
q
V + I ⋅ Rs
kT
−1 − I o2 e
q
V + I ⋅ Rs
2 kT
−1 −
V + I ⋅ Rs
Rp
(2.8)
It is possible to combine the first diode (D1) and the second diode (D2) and rewrite
the equation (2.8) in the following form.
I = I sc − I o e
q
V + I ⋅ Rs
nkT
−1 −
V + I ⋅ Rs
Rp
(2.9)
where: n is known as the “ideality factor” (“n” is sometimes denoted as “A”) and takes the
value between one and two [7].
16
27.
2.4 Photovoltaic Module
A single PV cell produces an output voltage less than 1V, about 0.6V for crystallinesilicon (Si) cells, thus a number of PV cells are connected in series to archive a desired
output voltage. When seriesconnected cells are placed in a frame, it is called as a module.
Most of commercially available PV modules with crystallineSi cells have either 36 or 72
seriesconnected cells. A 36cell module provides a voltage suitable for charging a 12V
battery, and similarly a 72cell module is appropriate for a 24V battery. This is because most
of PV systems used to have backup batteries, however today many PV systems do not use
batteries; for example, gridtied systems. Furthermore, the advent of high efficiency DCDC
converters has alleviated the need for modules with specific voltages. When the PV cells are
wired together in series, the current output is the same as the single cell, but the voltage
output is the sum of each cell voltage, as shown in Figure 27.
5
4.5
4
Current (A)
3.5
3
9 cells
36 cells
2.5
72 cells
3 cells
2
1.5
1
0.5
0
0
5
10
0.6V for each cell
15
20
25
Voltage (V)
30
35
40
45
Figure 27: PV cells are connected in series to make up a PV module
17
28.
Also, multiple modules can be wired together in series or parallel to deliver the
voltage and current level needed. The group of modules is called an array.
2.5 Modeling a PV Module by MATLAB
BP Solar BP SX 150S PV module, pictured in Figure 28, is chosen for a MATLAB
simulation model. The module is made of 72 multicrystalline silicon solar cells in series and
provides 150W of nominal maximum power [1]. Table 21 shows its electrical specification.
Figure 28: Picture of BP SX 150S PV module [1]
Electrical Characteristics
Maximum Power (Pmax)
Voltage at Pmax (Vmp)
Current at Pmax (Imp)
Opencircuit voltage (Voc)
Shortcircuit current (Isc)
Temperature coefficient of Isc
Temperature coefficient of Voc
Temperature coefficient of power
NOCT
150W
34.5V
4.35A
43.5V
4.75A
0.065 ± 0.015 %/ oC
160 ± 20 mV/ oC
0.5 ± 0.05 %/ oC
47 ± 2oC
Table 21: Electrical characteristics data of PV module taken from the datasheet [1]
The strategy of modeling a PV module is no different from modeling a PV cell. It
uses the same PV cell model. The parameters are the all same, but only a voltage parameter
(such as the opencircuit voltage) is different and must be divided by the number of cells.
18
29.
The study done by Walker [27] of University of Queensland, Australia, uses the
electric model with moderate complexity, shown in Figure 29, and provides fairly accurate
results. The model consists of a current source (Isc), a diode (D), and a series resistance (Rs).
The effect of parallel resistance (Rp) is very small in a single module, thus the model does not
include it. To make a better model, it also includes temperature effects on the shortcircuit
current (Isc) and the reverse saturation current of diode (Io). It uses a single diode with the
diode ideality factor (n) set to achieve the best IV curve match.
Rs
+
Isc
D
V
Load
Figure 29: Equivalent circuit used in the MATLAB simulations
Since it does not include the effect of parallel resistance (Rp), letting Rp =
in the
equation (2.9) gives the equation [27] that describes the currentvoltage relationship of the
PV cell, and it is shown below.
I = I sc − I o e
q
V + I ⋅ Rs
nkT
−1
where: I is the cell current (the same as the module current),
V is the cell voltage = {module voltage} ÷ {# of cells in series},
T is the cell temperature in Kelvin (K).
19
(2.10)
30.
First, calculate the shortcircuit current (Isc) at a given cell temperature (T):
[
]
I sc T = I sc Tref ⋅ 1 + a(T − Tref )
(2.11)
where: Isc at Tref is given in the datasheet (measured under irradiance of 1000W/m2),
Tref is the reference temperature of PV cell in Kelvin (K), usually 298K (25oC),
a is the temperature coefficient of Isc in percent change per degree temperature also
given in the datasheet.
The shortcircuit current (Isc) is proportional to the intensity of irradiance, thus Isc at a
given irradiance (G) is:
I sc G =
G
I sc Go
Go
(2.12)
where: Go is the nominal value of irradiance, which is normally 1KW/m2.
The reverse saturation current of diode (Io) at the reference temperature (Tref) is given
by the equation (2.6) with the diode ideality factor added:
Io =
I
(e
sc
qVoc / nkT
− 1)
(2.13)
The reverse saturation current (Io) is temperature dependant and the Io at a given
temperature (T) is calculated by the following equation [27].
I o T = I o Tref
T
⋅
Tref
3
n
⋅e
− q⋅ E g 1 1
−
n⋅k T Tref
(2.14)
The diode ideality factor (n) is unknown and must be estimated. It takes a value
between one and two; the value of n=1 (for the ideal diode) is, however, used until the more
accurate value is estimated later by curve fitting [27]. Figure 210 shows the effect of the
varying ideality factor.
20
31.
5
n=1
4.5
4
n=2
Module Current (A)
3.5
3
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
Figure 210: Effect of diode ideally factors by MATLAB simulation (1KW/m2, 25oC)
The series resistance (Rs) of the PV module has a large impact on the slope of the IV
curve near the opencircuit voltage (Voc), as shown in Figure 211, hence the value of Rs is
calculated by evaluating the slope
dV
of the IV curve at the Voc [27]. The equation for Rs is
dI
derived by differentiating the equation (2.10) and then rearranging it in terms of Rs.
I = I sc − I o e
q
V + I ⋅ Rs
nkT
−1
q
dV + Rs ⋅ dI
dI = 0 − I o ⋅ q
⋅e
nkT
Rs = −
dI
−
dV
(2.15)
V + I ⋅ Rs
nkT
(2.16)
nkT q
Io ⋅ e
q
V + I ⋅ Rs
nkT
21
(2.17)
32.
Then, evaluate the equation (2.17) at the open circuit voltage that is V=Voc (also let I=0).
Rs = −
where:
dV
dI
dV
dI
−
Voc
nkT q
Io ⋅ e
(2.18)
qVoc
nkT
is the slope of the IV curve at the Voc (use the IV curve in the datasheet then
Voc
divide it by the number of cells in series),
Voc is the opencircuit voltage of cell (found by dividing Voc in the datasheet by the
number of cells in series).
The calculation using the slope measurement of the IV curve published on the BP SX
150 datasheet gives a value of the series resistance per cell, Rs = 5.1m .
5
4.5
4
Rs=0
Module Current (A)
3.5
3
2.5
Rs=5 mOhm
2
Rs=10 mOhm
1.5
Rs=15 mOhm
1
0.5
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
Figure 211: Effect of series resistances by MATLAB simulation (1KW/m2, 25oC)
Finally, it is possible to solve the equation of IV characteristics (2.10). It is, however,
complex because the solution of current is recursive by inclusion of a series resistance in the
22
33.
model. Although it may be possible to find the answer by simple iterations, the Newton’s
method is chosen for rapid convergence of the answer [27]. The Newton’s method is
described as:
f (xn )
f ′( x n )
x n +1 = x n −
(2.19)
where: f ′(x ) is the derivative of the function, f ( x ) = 0 , x n is a present value, and x n +1 is a
next value.
Rewriting the equation (2.10) gives the following function:
f ( I ) = I sc − I − I o e
q
V + I ⋅ Rs
nkT
−1 = 0
(2.20)
Plugging this into the equation (2.19) gives a following recursive equation, and the output
current (I) is computed iteratively.
I sc − I n − I o e
I n +1 = I n −
−1− Io
q
V + I n ⋅ Rs
nkT
q ⋅ Rs
e
nkT
q
−1
V + I n ⋅ Rs
nkT
(2.21)
The MATLAB function written in this thesis performs the calculation five times
iteratively to ensure convergence of the results. The testing result has shown that the value
of In usually converges within three iterations and never more than four interactions. Please
refer to Appendix A.1.1 for this MATLAB function.
Figure 212 shows the plots of IV characteristics at various module temperatures
simulated with the MATLAB model for BP SX 150S PV module. Data points superimposed
on the plots are taken from the IV curves published on the manufacturer’s datasheet [1].
After some trials with various diode ideality factors, the MATLAB model chooses the value
23
34.
of n = 1.62 that attains the best match with the IV curve on the datasheet. The figure shows
good correspondence between the data points and the simulated IV curves.
750C
250C
0
50 C
O 0C
Figure 212: IV curves of BP SX 150S PV module at various temperatures
Simulated with the MATLAB model (1KW/m2, 25oC)
24
35.
2.6 The IV Curve and Maximum Power Point
Figure 213 shows the IV curve of the BP SX 150S PV module simulated with the
MATLAB model. A PV module can produce the power at a point, called an operating point,
anywhere on the IV curve. The coordinates of the operating point are the operating voltage
and current. There is a unique point near the knee of the IV curve, called a maximum power
point (MPP), at which the module operates with the maximum efficiency and produces the
maximum output power. It is possible to visualize the location of the by fitting the largest
possible rectangle inside of the IV curve, and its area equal to the output power which is a
product of voltage and current.
5
Isc = 4.75A
P3 = 94.9W
P1 = 150.0W
4.5
Impp = 4.35A
4
Maximum Power Point (MPP)
Module Current (A)
3.5
P2 = 108.2W
3
2.5
2
1.5
1
0.5
0
Vmpp = 34.5V
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Voc = 43.5V
Figure 213: Simulated IV curve of BP SX 150S PV module (1KW/m2, 25oC)
The power vs. voltage plot is overlaid on the IV plot of the PV module, as shown in
Figure 214. It reveals that the amount of power produced by the PV module varies greatly
25
36.
depending on its operating condition. It is important to operate the system at the MPP of PV
module in order to exploit the maximum power from the module. The next chapter will
discuss how to do it.
160
7
Pmax
140
6
120
Module Current (A)
Isc
5
100
4
Impp
80
MPP
3
60
2
40
1
0
Vmpp
0
5
10
15
20
25
30
Module Voltage (V)
Voc
35
40
Figure 214: IV and PV relationships of BP SX 150S PV module
Simulated with the MATLAB model (1KW/m2, 25oC)
26
20
0
45
Module Output Power (W)
8
37.
Chapter 3 Maximum Power Point Tracker
3.1 Introduction
When a PV module is directly coupled to a load, the PV module’s operating point
will be at the intersection of its I–V curve and the load line which is the IV relationship of
load. For example in Figure 31, a resistive load has a straight line with a slope of 1/Rload as
shown in Figure 32. In other words, the impedance of load dictates the operating condition
of the PV module. In general, this operating point is seldom at the PV module’s MPP, thus it
is not producing the maximum power. A study shows that a directcoupled system utilizes a
mere 31% of the PV capacity [11]. A PV array is usually oversized to compensate for a low
power yield during winter months. This mismatching between a PV module and a load
requires further oversizing of the PV array and thus increases the overall system cost. To
mitigate this problem, a maximum power point tracker (MPPT) can be used to maintain the
PV module’s operating point at the MPP. MPPTs can extract more than 97% of the PV
power when properly optimized [9].
This chapter discusses the IV characteristics of PV modules and loads, matching
between the two, and the use of DCDC converters as a means of MPPT. It also discusses
the details of some MPPT algorithms and control methods, and limitations of MPPT.
+
PV
I
V
R
Figure 31: PV module is directly connected to a (variable) resistive load
27
38.
5
*
4.5
R=4 Ohms
4
R=7.93 Ohms
*
MPP
Module Current (A)
3.5
3
2.5
Slope=1/R
*
R=16 Ohms
2
1.5
1
Increasing R
0.5
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Figure 32: IV curves of BP SX 150S PV module and various resistive loads
Simulated with the MATLAB model (1KW/m2, 25oC)
3.2 IV Characteristics of DC Motors
Many PV water pumping systems employ DC motors (instead of AC motors) because
they could be directly coupled with PV arrays and make a very simple system. Among
different types of DC motors, a permanent magnet DC (PMDC) motor is preferred in PV
systems because it can provide higher starting torque. Figure 33 shows an electrical model
of a PMDC motor. When the motor is turning, it produces a back emf, or a counterelectromotive force, described as an electric potential (E) proportional to the angular speed
( ) of the rotor. From the equivalent circuit, the DC voltage equation for the armature circuit
is:
V = I ⋅ Ra + K ⋅ ω
where: Ra is the armature resistance.
28
(3.1)
39.
The back emf is E=K·
where: K is the constant, and
is the angular speed of rotor in
rad/sec.
Ra
+
PV
I
E=Kw
V

Figure 33: Electrical model of permanent magnet DC motor
Figure 34 shows an example of currentvoltage relationship (IV curve) of a DC
motor. Applying the voltage to start the motor, the current rises rapidly with increasing
voltage until the current is sufficient to create enough starting torque to break the motor loose
from static friction [16]. At startup ( =0), there is no effect of back emf, therefore the
starting current builds up linearly with a steep slope of 1/Ra on the IV plot as shown in
Figure 34. Once it starts to run, the back emf takes effect and drops the current, therefore
the current rises slowly with increasing voltage.
As mentioned already a simple type of PV water pumping systems uses a direct
coupled PVmotor setup. This configuration has a severe disadvantage in efficiency because
of a mismatched operating point, as shown in Figure 34. For this example, the water
pumping system would not start operating until irradiance reaches at 400W/m2. Once it starts
to run, it requires as little as 200W/m2 of irradiance to maintain the minimum operation. This
means that the system cannot utilize a fair amount of morning insolation just because there is
insufficient starting torque. Also, when the motor is operated under the locked condition for
29
40.
a long time, it may result in shortening of the life of the motor due to input electrical energy
converted to heat rather than to mechanical output [15].
DC Motor IV Curve
1000W/m2
Slope = 1/Ra
Current
800W/m2
600W/m2
400W/m2
200W/m2
Voltage
Figure 34: PV IV curves with varying irradiance and a DC motor IV curve
There is a MPPT specifically called a linear current booster (LCB) that is designed to
overcome the above mentioned problem in water pumping systems. The MPPT maintains
the input voltage and current of LCB at the MPP of PV module. As shown in Figure 35, the
power produced at the MPP is relatively lowcurrent and highvoltage which is opposite of
those required by the pump motor. The LCB shifts this relationship around and converts into
highcurrent and lowvoltage power which satisfies the pump motor characteristics. For the
example in Figure 35, tracing of the isopower (constant power) line from the MPP reveals
that the LCB could start the pump motor with as little as 50W/m2 of irradiance (assuming the
LCB can convert the power without loss).
30
41.
DC Motor IV Curve
1000W/m2
MPP
Isopower line
800W/m2
Current
600W/m2
400W/m2
200W/m2
50W/m2
Voltage
Figure 35: PV IV curves with isopower lines (dotted) and a DC motor IV curve
3.3 DCDC Converter
The heart of MPPT hardware is a switchmode DCDC converter. It is widely used in
DC power supplies and DC motor drives for the purpose of converting unregulated DC input
into a controlled DC output at a desired voltage level [17]. MPPT uses the same converter
for a different purpose: regulating the input voltage at the PV MPP and providing loadmatching for the maximum power transfer.
3.3.1 Topologies
There are a number of different topologies for DCDC converters.
They are
categorized into isolated or nonisolated topologies.
The isolated topologies use a smallsized highfrequency electrical isolation
transformer which provides the benefits of DC isolation between input and output, and step
31
42.
up or down of output voltage by changing the transformer turns ratio. They are very often
used in switchmode DC power supplies [18]. Popular topologies for a majority of the
applications are flyback, halfbridge, and fullbridge [22]. In PV applications, the gridtied
systems often use these types of topologies when electrical isolation is preferred for safety
reasons.
Nonisolated topologies do not have isolation transformers. They are almost always
used in DC motor drives [17]. These topologies are further categorized into three types: step
down (buck), step up (boost), and step up & down (buckboost). The buck topology is used
for voltage stepdown. In PV applications, the buck type converter is usually used for
charging batteries and in LCB for water pumping systems. The boost topology is used for
stepping up the voltage. The gridtied systems use a boost type converter to step up the
output voltage to the utility level before the inverter stage. Then, there are topologies able to
step up and down the voltage such as: buckboost, Cúk, and SEPIC (stands for Single Ended
Primary Inductor Converter). For PV system with batteries, the MPP of commercial PV
module is set above the charging voltage of batteries for most combinations of irradiance and
temperature. A buck converter can operate at the MPP under most conditions, but it cannot
do so when the MPP goes below the battery charging voltage under a lowirradiance and
hightemperature condition. Thus, the additional boost capability can slightly increase the
overall efficiency [27].
3.3.2 Cúk and SEPIC Converters
For water pumping systems, the output voltage needs to be stepped down to provide a
higher starting current for a pump motor. The buck converter is the simplest topology and
easiest to understand and design, however it exhibits the most severe destructive failure mode
32
43.
of all configurations [22]. Another disadvantage is that the input current is discontinuous
because of the switch located at the input, thus good input filter design is essential. Other
topologies capable of voltage stepdown are Cúk and SEPIC. Even though their voltage
stepup function is optional for LCB application, they have several advantages over the buck
converter. They provide capacitive isolation which protects against switch failure (unlike the
buck topology) [21]. The input current of the Cúk and SEPIC topologies is continuous, and
they can draw a ripple free current from a PV array that is important for efficient MPPT.
Figure 36 shows a circuit diagram of the basic Cúk converter. It is named after its
inventor. It can provide the output voltage that is higher or lower than the input voltage. The
SEPIC, a derivative of the Cúk converter, is also able to step up and down the voltage.
Figure 37 shows a circuit diagram of the basic SEPIC converter. The characteristics of two
topologies are very similar. They both use a capacitor as the main energy storage. As a
result, the input current is continuous. The circuits have low switching losses and high
efficiency [18]. The main difference is that the Cúk converter has a polarity of the output
voltage reverse to the input voltage. The input and output of SEPIC converter have the same
voltage polarity; therefore the SEPIC topology is sometimes preferred to the Cúk topology.
SEPIC maybe also preferred for battery charging systems because the diode placed on the
output stage works as a blocking diode preventing an adverse current going to PV source
from the battery. The same diode, however, gives the disadvantage of highripple output
current.
On the other hand, the Cúk converter can provide a better output current
characteristic due to the inductor on the output stage. Therefore, the thesis decides on the
Cúk converter because of the good input and output current characteristics.
33
44.
Figure 36: Circuit diagram of the basic Cúk converter
Figure 37: Circuit diagram of the basic SEPIC converter
3.3.3 Basic Operation of Cúk Converter
The basic operation of Cúk converter in continuous conduction mode is explained
here. In steady state, the average inductor voltages are zero, thus by applying Kirchoff’s
voltage law (KVL) around outermost loop of the circuit shown in Figure 36 [21].
VC 1 = V s + V o
(3.2)
Assume the capacitor (C1) is large enough and its voltage is ripple free even though it stores
and transfer large amount of energy from input to output [17] (this requires a good low ESR
capacitor [21]).
The initial condition is when the input voltage is turned on and switch (SW) is off.
The diode (D) is forward biased, and the capacitor (C1) is being charged. The operation of
circuit can be divided into two modes.
34
45.
Mode 1: When SW turns ON, the circuit becomes one shown in Figure 38.
Figure 38: Basic Cúk converter when the switch is ON
The voltage of the capacitor (C1) makes the diode (D) reversebiased and turned off.
The capacitor (C1) discharge its energy to the load through the loop formed with SW, C2, Rload,
and L2. The inductors are large enough, so assume that their currents are ripple free. Thus,
the following relationship is established [21].
− I C1 = I L 2
(3.3)
Mode 2: When SW turns OFF, the circuit becomes one shown in Figure 39.
Figure 39: Basic Cúk converter when the switch is OFF
The capacitor (C1) is getting charged by the input (Vs) through the inductor (L1). The
energy stored in the inductor (L2) is transfer to the load through the loop formed by D, C2,
and Rload. Thus, the following relationship is established [21].
I C 1 = I L1
(3.4)
35
46.
For periodic operation, the average capacitor current is zero. Thus, from the equation (3.3)
and (3.4) [21]:
[I
C 1 SW
ON
]⋅ DT + [I
C 1 SW
OFF
]⋅ (1 − D )T = 0
(3.5)
− I L 2 ⋅ DT + I L1 ⋅ (1 − D )T = 0
(3.6)
I L1
D
=
I L2 1 − D
(3.7)
where: D is the duty cycle (0 < D < 1), and T is the switching period.
Assuming that this is an ideal converter, the average power supplied by the source
must be the same as the average power absorbed by the load [21].
Pin = Pout
(3.8)
V s ⋅ I L1 = V o ⋅ I L 2
(3.9)
I L1 Vo
=
I L 2 Vs
(3.10)
Combining the equation (3.7) and (3.10), the following voltage transfer function is derived
[21].
Vo
D
=
Vs 1 − D
(3.11)
Its relationship to the duty cycle (D) is:
If 0 < D < 0.5 the output is smaller than the input.
If D = 0.5 the output is the same as the input.
If 0.5 < D < 1 the output is larger than the input.
36
47.
3.4 Mechanism of Load Matching
As described in Section 3.1, when PV is directly coupled with a load, the operating
point of PV is dictated by the load (or impedance to be specific). The impedance of load is
described as below.
Rload =
Vo
Io
(3.12)
where: Vo is the output voltage, and Io is the output current.
The optimal load for PV is described as:
Ropt =
V MPP
I MPP
(3.13)
where: VMPP and IMPP are the voltage and current at the MPP respectively. When the value of
Rload matches with that of Ropt, the maximum power transfer from PV to the load will occur.
These two are, however, independent and rarely matches in practice. The goal of the MPPT
is to match the impedance of load to the optimal impedance of PV.
The following is an example of load matching using an ideal (lossless) Cúk
converter. From the equation (3.11):
1− D
⋅ Vo
D
(3.14)
Is
V
I
= L1 = o
I o I L 2 Vs
(3.15)
Vs =
From the equation (3.10),
From the equation (3.14) and (3.15),
Is =
D
⋅ Io
1− D
37
(3.16)
48.
From the equation (3.14) and (3.16), the input impedance of the converter is:
V s (1 − D ) 2 Vo (1 − D ) 2
Rin =
=
⋅
=
⋅ Rload
Is
Io
D2
D2
(3.17)
As shown in Figure 310, the impedance seem by PV is the input impedance of the
converter (Rin). By changing the duty cycle (D), the value of Rin can be matched with that of
Ropt. Therefore, the impedance of the load can be anything as long as the duty cycle is
adjusted accordingly.
+
PV
Rin
DCDC Conv
Rload
Figure 310: The impedance seen by PV is Rin that is adjustable by duty cycle (D)
3.5 Maximum Power Point Tracking Algorithms
The location of the MPP in the I–V plane is not known beforehand and always
changes dynamically depending on irradiance and temperature. For example, Figure 311
shows a set of PV I–V curves under increasing irradiance at the constant temperature (25oC),
and Figure 312 shows the I–V curves at the same irradiance values but with a higher
temperature (50oC). There are observable voltage shifts where the MPP occurs. Therefore,
the MPP needs to be located by tracking algorithm, which is the heart of MPPT controller.
There are a number of methods that have been proposed. One method measures an
opencircuit voltage (Voc) of PV module every 30 seconds by disconnecting it from rest of the
circuit for a short moment. Then, after reconnection, the module voltage is adjusted to 76%
38
49.
of measured Voc which corresponds to the voltage at the MPP [6] (note: the percentage
depends on the type of cell used). The implementation of this openloop control method is
very simple and lowcost although the MPPT efficiencies are relatively low (between
73~91%) [9]. Model calculations can also predict the location of MPP; however in practice
it does not work well because it does not take physical variations and aging of module and
other effects such as shading into account.
Furthermore, a pyranometer that measures
irradiance is quite expensive. Search algorithm using a closedloop control can achieve
higher efficiencies, thus it is the customary choice for MPPT. Among different algorithms,
the Perturb & Observe (P&O) and Incremental Conductance (incCond) methods are studied
here.
5
1000W/m2
Maximum Power Point
4.5
4
750W/m2
Module Current (A)
3.5
3
500W/m2
2.5
2
1.5
250W/m2
1
0.5
0
50W/m2
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Figure 311: IV curves for varying irradiance and a trace of MPPs (25oC)
39
50.
5
1000W/m2
Maximum Power Point
4.5
4
750W/m2
Module Current (A)
3.5
3
500W/m2
2.5
2
1.5
250W/m2
1
0.5
0
50W/m2
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Figure 312: IV curves for varying irradiance and a trace of MPPs (50oC)
3.5.1 Perturb & Observe Algorithm
The perturb & observe (P&O) algorithm, also known as the “hill climbing” method,
is very popular and the most commonly used in practice because of its simplicity in
algorithm and the ease of implementation. The most basic form of the P&O algorithm
operates as follows. Figure 313 shows a PV module’s output power curve as a function of
voltage (PV curve), at the constant irradiance and the constant module temperature,
assuming the PV module is operating at a point which is away from the MPP. In this
algorithm the operating voltage of the PV module is perturbed by a small increment, and the
resulting change of power, P, is observed. If the P is positive, then it is supposed that it
has moved the operating point closer to the MPP. Thus, further voltage perturbations in the
same direction should move the operating point toward the MPP. If the P is negative, the
40
51.
operating point has moved away from the MPP, and the direction of perturbation should be
reversed to move back toward the MPP. Figure 314 shows the flowchart of this algorithm.
160
MPP
140
Module Output Power (W)
120
100
A
80
*
*
B
60
40
20
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Figure 313: Plot of power vs. voltage for BP SX 150S PV module (1KW/m2, 25oC)
Figure 314: Flowchart of the P&O algorithm
41
52.
There are some limitations that reduce its MPPT efficiency. First, it cannot determine
when it has actually reached the MPP. Instead, it oscillates the operating point around the
MPP after each cycle and slightly reduces PV efficiency under the constant irradiance
condition [9]. Second, it has been shown that it can exhibit erratic behavior in cases of
rapidly changing atmospheric conditions as a result of moving clouds [11]. The cause of this
problem can be explained using Figure 315 with a set of PV curves with varying irradiance.
Assume that the operating point is initially at the point A and is oscillating around the MPP
at the irradiance of 250W/m2. Then, the irradiance increases rapidly to 500W/m2. The power
measurement results in a positive P. If this operating point is perturbing from right to left
around the MPP, then the operating point will actually moves from the point A toward the
point E (instead of B). This happens because the MPPT can not tell that the positive P is
the result of increasing irradiation and simply assumes that it is the result of moving the
operating point to closer to the MPP. In this case the positive P is measured when the
operating voltage has been moving toward the left; the MPPT is fooled as if there is a MPP
on the left side. If the irradiance is still rapidly increasing, again the MPPT will see the
positive P and will assume it is moving towards the MPP, continuing to perturb to the left.
From points A, E, F and G, the operating point continues to deviate from the actual MPP
until the solar radiation change slows or settles down. This situation can occur on partly
cloudy days, and MPP tracking is most difficult because of the frequent movement of the
MPP.
42
54.
days because the power change due to irradiance makes the step size too big. A modification
involving taking a PV power measurement twice at the same voltage solves the problem of
not detecting the changing irradiance [9]. Comparing these two measurements, the algorithm
can determine whether the irradiance is changing and decide how to perturb the operating
point. The tradeoff is that the increased number of sampling slows response times and
increases the complexity of algorithm.
3.5.2 Incremental Conductance Algorithm
In 1993 Hussein, Muta, Hoshino, and Osakada of Saga University, Japan, proposed
the incremental conductance (incCond) algorithm intending to solve the problem of the PO
algorithm under rapidly changing atmospheric conditions [11].
The basic idea is that the slope of PV curve becomes zero at the MPP, as shown in
Figure 313. It is also possible to find a relative location of the operating point to the MPP
by looking at the slopes. The slope is the derivative of the PV module’s power with respect
to its voltage and has the following relationships with the MPP.
dP
= 0 at MPP
dV
(3.18)
dP
0 at the left of MPP
dV
(3.19)
dP
0 at the right of MPP
dV
(3.20)
The above equations are written in terms of voltage and current as follows.
dP d (V ⋅ I )
dV
dI
dI
=
=I
+V
= I +V
dV
dV
dV
dV
dV
44
(3.21)
55.
If the operating point is at the MPP, the equation (3.21) becomes:
I +V
dI
=0
dV
(3.22)
dI
I
=−
dV
V
(3.23)
If the operating point is at the left side of the MPP, the equation (3.21) becomes:
I +V
dI
0
dV
(3.24)
dI
I
−
dV
V
(3.25)
If the operating point is at the right side of the MPP, the equation (3.21) becomes:
I +V
dI
0
dV
(3.26)
dI
I
−
dV
V
(3.27)
Note that the left side of the equations (3.23), (3.25), and (3.27) represents
incremental conductance of the PV module, and the right side of the equations represents its
instantaneous conductance.
The flowchart shown in Figure 316 explains the operation of this algorithm. It starts
with measuring the present values of PV module voltage and current. Then, it calculates the
incremental changes, dI and dV, using the present values and previous values of voltage and
current. The main check is carried out using the relationships in the equations (3.23), (3.25),
and (3.27). If the condition satisfies the inequality (3.25), it is assumed that the operating
point is at the left side of the MPP thus must be moved to the right by increasing the module
voltage. Similarly, if the condition satisfies the inequality (3.27), it is assumed that the
operating point is at the right side of the MPP, thus must be moved to the left by decreasing
45
56.
the module voltage. When the operating point reaches at the MPP, the condition satisfies the
equation (3.23), and the algorithm bypasses the voltage adjustment. At the end of cycle, it
updates the history by storing the voltage and current data that will be used as previous
values in the next cycle. Another important check included in this algorithm is to detect
atmospheric conditions. If the MPPT is still operating at the MPP (condition: dV = 0) and
the irradiation has not changed (condition: dI = 0), it takes no action. If the irradiation has
increased (condition: dI 0), it raises the MPP voltage. Then, the algorithm will increase the
operating voltage to track the MPP. Similarly, if the irradiation has decreased (condition: dI
0), it lowers the MPP voltage. Then, the algorithm will decrease the operating voltage.
dI
I
=−
dV
V
dI
I
−
dV
V
Figure 316: Flowchart of the incCond algorithm
46
57.
In practice, the condition dP/dV = 0 (or dI/dV = I/V) seldom occurs because of the
approximation made in the calculation of dI and dV [11]. Thus, a small margin of error (E)
should be allowed, for example: dP/dV = ±E. The value of E is optimized with exchange
between an amount of the steadysate tracking error and a risk of oscillation of the operating
point.
3.6 Control of MPPT
As explained in the previous section, the MPPT algorithm tells a MPPT controller
how to move the operating voltage. Then, it is a MPPT controller’s task to bring the voltage
to a desired level and maintain it. There are several methods often used for MPPT.
3.6.1 PI Control
As shown in Figure 317, the MPPT takes measurement of PV voltage and current,
and then tracking algorithm (PO, incCond, or variations of two) calculates the reference
voltage (Vref) where the PV operating voltage should move next.
The task of MPPT
algorithm is to set Vref only, and it is repeated periodically with a slower rate (typically 1~10
samples per second). Then, there is another control loop that the proportional and integral
(PI) controller regulates the input voltage of converter. Its task is to minimize error between
Vref and the measured voltage by adjusting the duty cycle. The PI loop operates with a much
faster rate and provides fast response and overall system stability [10] [12]. The PI controller
itself can be implemented with analog components, but it is often done with DSPbased
controller [10] because the DSP can handle other tasks such as MPP tracking thus reducing
parts count.
47
58.
Figure 317: Block diagram of MPPT with the PI compensator
3.6.2 Direct Control
As shown in Figure 318, this control method is simpler and uses only one control
loop, and it performs the adjustment of duty cycle within the MPP tracking algorithm. The
way how to adjust the duty cycle is totally based on the theory of load matching explained in
Section 3.4.
Figure 318: Block diagram of MPPT with the direct control
48
59.
The impedance seen by PV is the input impedance of converter. Using the example of the
Cúk converter in Section 3.4, the relationship to the load is:
V s (1 − D ) 2
Rin =
=
⋅ Rload
Is
D2
(3.28)
where: D is the duty cycle of the Cúk converter. As shown in Figure 319, increasing D will
decrease the input impedance (Rin), thus the PV operating voltage moves to the left.
Similarly, decreasing D will increase Rin, thus the operating voltage moves to the right. The
tracking algorithm (PO, incCond, or variations of two) makes the decision how to move the
operating voltage.
5
*
4.5
R=4 Ohms
4
R=7.93 Ohms
*
MPP
Module Current (A)
3.5
3
2.5
Slope=1/R
*
R=16 Ohms
2
Increasing D
1.5
Increasing Rin
1
0.5
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Figure 319: Relationship of the input impedance of Cúk converter and its duty cycle
The time response of the power stage and PV source is relatively slow (10~50msec
depending on the type of load) [9]. The MPPT algorithm changes the duty cycle, then the
next sampling of PV voltage and current should be taken after the system reaches the
periodic steady state to avoid measuring the transient behavior [9]. The typical sampling rate
49
60.
is 10~100 samples per second. The sampling rate of PI controller is much faster, thus it
provides robustness against sudden changes of load. The system response is, however, slow
in general. The direct control method can operate stably for applications such as battery
equipped systems and water pumping systems. Since sampling rates are slow, it is possible
to implement with inexpensive microcontrollers [12].
3.6.3 Output Sensing Direct Control
This method is a variation of the aforementioned direct control and has the advantage
of requiring only two sensors for output voltage and current. The aforementioned two
methods use input sensing which enables accurate control of module’s operating point. In
addition with input sensors, however, they usually require another set of sensors for the
output to detect the overvoltage and overcurrent condition of load. The requirement of four
sensors often makes difficult to allow for low cost systems.
This output sensing method measures the power change of PV at the output side of
converter and uses the duty cycle as a control variable. The following MATLAB simulation
illustrates the relationship between the output power of converter and its duty cycle. In the
simulation, BP SX 150S PV module is coupled with the ideal (lossless) Cúk converter with
a resistive load (6 ). The duty cycle of converter is swept from 0 to 1 with 1% step, and the
output power of converter is plotted in Figure 320.
50
61.
160
140
Output Power (W)
120
100
80
60
40
20
0
0
0.1
0.2
0.3
0.4
0.5
0.6
Duty Cycle
0.7
0.8
0.9
1
Figure 320: Output power of Cúk converter vs. its duty cycle (1KW/m2, 25oC)
As shown in the figure, there is a peak of output power when the duty cycle of
converter is varied. This control method employs the PO algorithm to locate the MPP.
Figure 321 shows the flowchart of algorithm. In order to accommodate duty cycle as a
control variable, the PO algorithm used here is a slightly modified version from that
previously introduced, but the idea how it works is the same. The algorithm perturbs the
duty cycle and measure the output power of converter. If the power is increased, the duty
cycle is further perturbed in the same direction; otherwise the direction will be reversed.
When the output power of converter is reached at the peak, a PV module or array is supposed
to be operating at the MPP.
Even though it works perfectly in the simulation with the ideal converter, there is
some uncertainly if the peak of output power is corresponding with the MPP in practice with
51
62.
nonideal converters. Also, this control method only works with the PO algorithm and its
variations, and it does not work with the incCond algorithm.
Figure 321: Flowchart of PO algorithm for the output sensing direct control method
3.7 Limitations of MPPT
The main drawback of MPPT is that there is no regulation on output while it is
tracking a maximum power point. It cannot regulate both input and output at the same time.
The example of load matching in Section 3.4 is elaborated here to show how the output
voltage and current change with varying irradiation.
The maximum power transfer occurs when the input impedance of converter matches
with the optimal impedance of PV module, as described in the equation below.
Rin = Ropt =
V MPP
I MPP
52
(3.29)
63.
The equation (3.17) for the Cúk converter is solved for duty cycle (D).
1
D=
1+
Rin
Rload
(3.30)
From the equation (3.11), the output voltage of converter is:
Vo =
D
⋅ Vs
1− D
(3.31)
From the equation (3.16), the output current of converter is:
Io =
1− D
⋅ Is
D
(3.32)
The calculation results are tabulated in the tables below.
PV module data are
obtained from the MATLAB simulation model. Using the equations above, two sets of data
are collected for the resistive load of 6
Irradiance
1000W/m2
800W/m2
600W/m2
400W/m2
200W/m2
VMPP
34.5V
34.1V
33.6V
32.7V
31.1V
PV Module
IMPP
4.35A
3.48A
2.61A
1.73A
0.87A
and 12 at the constant module temperature of 25oC.
Pmax
150.0W
118.8W
87.7W
56.9W
26.9W
Rin
7.92
9.80
12.9
18.8
35.9
D
0.465
0.439
0.406
0.361
0.290
MPPT
Vo
30.0V
26.7V
22.9V
18.5.V
12.7V
Io
5.00A
4.45A
3.82A
3.08A
2.12A
Rload
6
6
6
6
6
Table 31: Load matching with the resistive load (6 ) under the varying irradiance
Irradiance
1000W/m2
800W/m2
600W/m2
400W/m2
200W/m2
VMPP
34.5V
34.1V
33.6V
32.7V
31.1V
PV Module
IMPP
4.35A
3.48A
2.61A
1.73A
0.87A
Pmax
150.0W
118.8W
87.7W
56.9W
26.9W
Rin
7.92
9.80
12.9
18.8
35.9
D
0.552
0.525
0.491
0.444
0.366
MPPT
Vo
42.4V
37.8V
32.4V
26.1V
18.0V
Io
3.54A
3.15A
2.70A
2.18A
1.50A
Table 32: Load matching with the resistive load (12 ) under the varying irradiance
53
Rload
12
12
12
12
12
64.
From the above results, it’s obvious that there is no regulation of the output voltage
and current. If the application requires a constant voltage, it must employ batteries to
maintain the voltage constant. For water pumping system without batteries, the lack of
output regulation is not a predicament as long as they are equipped with water reservoirs to
meet the demand of water. The speed of pump motor is proportional to the converter’s
output voltage which is relative to irradiation. Thus, when the sun shines more, it simply
pumps more water.
Another noteworthy fact is that MPPT stops its original task if the load cannot
consume all the power delivered. For the standalone system, when the load is limited by its
maximum voltage or current, the MPPT moves the operating point away from the MPP and
sends less power. It is very important to select an appropriate size of load, thus it can utilize
the full capacity of PV module and array. On the other hand, the gridtied system can always
perform the maximum power point tracking because it can inject the power into the grid as
much as produced.
Of course, in reality DCDC converter used in MPPT is not 100% efficient. The
efficiency gain from MPPT is large, but the system needs to take efficiency loss by DCDC
converter into account. There is also tradeoff between efficiency and the cost. It is necessary
for PV system engineers to perform economic analysis of different systems and also
necessary to seek other methods of efficiency improvement such as the use of a sun tracker.
After due consideration of limitations, the next chapter will discuss designs and
simulations of MPPT and PV water pumping system.
54
65.
Chapter 4 Design and Simulations
4.1 Introduction
This chapter provides the design and simulations of MPPT.
It discusses Cúk
converter design. After the component selection, PSpice simulations validate the design and
choice of the MPPT sampling rate. MATLAB simulations perform comparative tests of the
PO and incCond algorithm. Simulations also verify the functionality of MPPT with a
resistive load and then with the DC pump motor load.
At last, this chapter provides
comparisons between the PV water pumping system equipped with MPPT and the directcoupled system without MPPT.
4.2 Cúk Converter Design
The basic operation of Cúk Converter and derivation of the voltage transfer function
is explained in Section 3.3.3. Here, a Cúk converter is designed based on the specification
shown in the table below. After component selection, the design is simulated in PSpice.
Specification
Input Voltage (Vs)
Input Current (Is)
Output Voltage (Vo)
Output Current (Io)
Maximum Output Power (Pmax)
Switching Frequency (f)
Duty Cycle (D)
2048V
05A ( 5% ripple)
1230V ( 5% ripple)
05A ( 5% ripple)
150W
50KHz
0.1 D 0.6
Table 41: Design specification of the Cúk Converter
55
66.
4.2.1 Component Selection
a) Inductor Selection
The inductor sizes are decided such that the change in inductor currents is no more
than 5% of the average inductor current.
The following equation gives the change in
inductor current [8].
∆i L =
Vs ⋅ D
L⋅ f
(4.1)
where: Vs is the input voltage, D is the duty cycle, and f is the switching frequency.
Solving this for L gives:
L=
Vs ⋅ D
∆i L ⋅ f
(4.2)
Assume that the worst current ripple will occur under the maximum power condition. Under
this condition, the average current (IL1) of the input inductor (L1) is 4.35A, and the ripple
current is 5% of IL1.
∆i L1 = 0.05 ⋅ I L1 = (0.05)( 4.35) = 0.2175 A
(4.3)
Thus, from the equation (4.2):
L1 =
Vs ⋅ D
(34.5)(0.465)
=
= 1.475 mH
∆i L1 ⋅ f (0.2175 )(50 × 10 3 )
(4.4)
A commercially available 1.5mH inductor is selected. For example, 1.5mH power coke
(5.0A DC max, 0.07 DCR) is available from Hammond Mfg. (www.hammondmfg.com).
Similarly, the value of the output inductor (L2) is calculated as follows.
∆i L 2 = 0.05 ⋅ I L 2 = (0.05)(5.0) = 0.250 A
56
(4.5)
67.
L2 =
Vs ⋅ D
(34.5)(0.465)
=
= 1.283mH
∆i L 2 ⋅ f (0.25)(50 × 10 3 )
(4.6)
To make parts procurement easier, the output can use the same inductor size as one in the
input.
b) Capacitor Selection
The design criterion for capacitors is that the ripple voltage across them should be
less than 5%. The average voltage across the capacitor (C1) is, from the equation (3.2), Vc1 =
Vs + Vo= 34.5 + 30 = 64.5V, so the maximum ripple voltage is vC1 = 0.05 × 64.5 = 3.225V.
The equivalent load resistance is:
2
Vo
(30.0) 2
R=
=
= 6Ω
Po
(150)
(4.7)
The value of C1 is calculated with the following equation [8]:
C1 =
Vo ⋅ D
(30.0)(0.465)
=
= 14 .42 µF
R ⋅ f ⋅ ∆v c1 (6)(50 × 10 3 )(3.225)
(4.8)
The next commercially available size is 22 F. An aluminum electrolytic capacitor with low
ESR type is required.
The value of the output capacitor (C2) is calculated using the output voltage ripple
equation (the same as that of buck converter) [21].
∆v o
1− D
=
Vo
8 ⋅ L2 ⋅ C 2 ⋅ f
2
(4.9)
Solving the above equation for C2 gives:
C2 =
1− D
8 ⋅ ( ∆v o Vo ) ⋅ L 2 ⋅ f
2
=
1 − 0.465
= 0.3567 µF
8(0.05)(1.5 × 10 −3 )(50 × 10 3 ) 2
(4.10)
The next available size is 0.47 F. An aluminum electrolytic capacitor with low ESR type is
required.
57
68.
c) Diode Selection
Schottky diode should be selected because it has a low forward voltage and very good
reverse recovery time (typically 5 to 10ns) [21]. From Figure 38, the recurrent peak reverse
voltage (VRRM) of the diode is the same as the average voltage of capacitor (C1) [18], thus
VRRM = 64.5V. Adding the 30% of safety factor gives the voltage rating of 83.9V. The
average forward current (IF) of diode is the combination of input and output currents at the
SW off, thus it is ID = IL1+IL2 = 9.35A. Adding the 30% of safety factor gives the current
rating of 12.2A. Schottky diodes are widely available from numerous vendors. For example,
MBR15100 (IF=15Amax, VRRM=100Vmax) meets the abovementioned voltage and current
ratings.
d) Switch Selection
PowerMOSFETs are widely used for low to medium power applications. The peak
voltage of the switch (SW) [18] is obtained by KVL on the circuit shown in Figure 39.
VSW = Vs −
dI L1
dt
(4.11)
The voltage of SW could go up to 48V by the specification. Adding the 30% of safety factor
gives the voltage rating of 62.4V. The peak switch current is the same as the diode. Thus,
adding the 30% of safety factor gives the current rating of 12.2A. There are a wide variety of
PowerMOSFETs available from various vendors.
VDS=100Vmax) meets the abovementioned requirements.
58
For example, IRF530 (ID=14Amax,
69.
4.2.2 PSpice Simulations
PSpice simulations validate the Cúk converter designed in Section 4.2.1. Figure 41
shows the circuit diagram with the PMDC motor model as a load. In the diagram, Ra and La
are resistance and inductance of armature winding, respectively, and E is the back emf of the
motor. The converter is running with full load. The values of armature resistance and
inductance that correspond to the actual DC pump motor are unknown, thus they are
estimated from other references [2] [20]. A more detailed discussion of modeling a DC
motor appears in Section 4.5.1.
1
C1
L1
2
2
1.5mH
34.5Vdc
Vpwm
La
1
2
1.5mH
S1
Vin
L2
+

1
10mH
22uF
Ra
.2
+

D1
Dbreak
Sbreak
C2
.47uF
E
28Vdc
0
0
V1 = 0
V2 = 1
TR = 10n
TF = 10n
TD = 0
PW = {D*T}
PER = {T}
PARAMETERS:
f = 50kHz
T = {1/f }
D = .465
Figure 41: Schematic of the Cúk converter with PMDC motor load
Figure 42 shows current and voltage plots of the converter after turning on (t = 0sec).
Since the load has such a large inductance, it takes a long time for current to build up. The
plots show that both input and output currents take nearly 250msec to reach steady state.
59
70.
5.0A
2.5A
SEL
0A
I(L1)
I(L2)
40V
30V
20V
10V
0V
0s
50ms
V(Vin:+)
100ms
150ms
200ms
250ms
V(C2:2)
Time
Figure 42: PSpice plots of input/output current (above) and voltage (below)
For comparisons, the same simulation is done with an equivalent resistive load (6 ).
The transient time is less than 10msec with the resistive load. It is apparent that the motor
load has a very slow response. Other current and voltage data are gathered and tabulated
below for comparisons with the resistive load and calculated results.
DC Motor
1 Set
2nd Set
4.07A
4.18A
5.2%
6.1%
4.70A
4.84A
4.6%
4.6%
34.5V
34.5V
n/a
n/a
28.9V
29.1V
9%
3.1%
st
Iin
Iout
Vin
Vout
Average
% ripple
Average
% ripple
Average
% ripple
Average
% ripple
Resistive
Load (6 )
4.20A
5.1%
4.83A
4.4%
34.5V
n/a
29.0V
2.7%
Calculated
Results
4.35A
5%
5.0A
5%
34.5V
n/a
30V
5%
Table 42: Cúk converter design: comparisons of simulations and calculated results
Table 42 shows two sets of simulation data for the DC motor load. The first set is
the result of simulation using the components selected in the previous section. The output
60
71.
voltage ripple for the DC motor load is as large as 9% while one for the equivalent resistive
load (6 ) is only 2.7%. Therefore, in the next simulation, the size of output capacitor (C2) is
increased to the next commercially available size of 1 F. It makes the input current ripple
slightly worse, but it makes overall improvement of performance. Thus, a 1 F capacitor
(instead of 0.47 F) is finally selected.
4.2.3 Choice of MPPT Sampling Rate
MPPT algorithms adjust PV operating point with a small step. The size of step is
typically 0.5V or less. For the Cúk converter designed, 0.5V corresponds to approximately
0.35% change in duty cycle. PSpice performs the simulation when the duty cycle is changed,
and the transient responses of voltage and current are observed. The use of “Sw_tOpen” and
“Sw_tClose” in the analog miscellaneous library permits switching of one duty cycle to
another duty cycle during the simulation. In the same way, it is also necessary to adjust the
value of back emf (E) because it has to correspond with the change of output voltage.
5.0A
4.5A
4.0A
I(L1)
I(L2)
35.0V
32.5V
30.0V
SEL
27.0V
240ms
V(L1:1)
260ms
V(C2:2)
280ms
300ms
320ms
340ms
Time
Figure 43: Transient response when duty cycle is increased 0.35% at 250ms
61
360ms
72.
Figure 43 is the result of PSpice simulation. It shows both input and output currents
take between 80msec and 90msec to go to steady state, where they take only several
milliseconds for the resistive load. It is important for MPPT algorithm to take measurements
of voltage and current after they reach steady state. Therefore, with a PV pump motor, the
sampling rate is 10Hz at most.
4.3 Comparisons of PO and incCond Algorithm
The two MPPT algorithms, PO and incCond, discussed in Section 3.5 are
implemented in MATLAB simulations and tested for their performance. Since the purpose is
to make comparisons of two algorithms, each simulation contains only the PV model and the
algorithm in order to isolate any influence from a converter or load.
First, they are verified to locate the MPP correctly under the constant irradiance, as
shown in Figure 44. Please refer to Appendix A.1 for MATLAB scripts for this section.
160
end
140
Module Output Power (W)
120
100
start
80
60
40
20
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
Figure 44: Searching the MPP (1KW/m2, 25oC)
The traces of PV operating point are shown in green, and the MPP is the red asterisk
62
73.
Next, the algorithms are tested with actual irradiance data provided by [2].
Simulations use two sets of data, shown in Figure 45; the first set of data is the
measurements of a sunny day in April in Barcelona, Spain, and the second set of data is for a
cloudy day in the same month at the same location.
The data contain the irradiance
measurements taken every two minutes for 12 hours. Irradiance values between two data
points are estimated by the cubic interpolation in MATLAB functions.
Sunny Day
Cloudy Day
1
Irradiance (KW/m2)
0.8
0.6
0.4
0.2
0
0
2
4
6
Hour (h)
8
10
12
Figure 45: Irradiance data for a sunny and a cloudy day of April in Barcelona, Spain [2]
On a sunny day, the irradiance level changes gradually since there is no influence of
cloud. MPP tracking is supposed to be easy. As shown in Figure 46, both algorithms locate
and maintain the PV operating point very close to the MPPs (shown in red asterisks) without
much difference in their performance.
63
74.
(a) PO algorithm
160
1000W/m
140
1000W/m2
140
800W/m2
100
600W/m2
80
400W/m2
60
800W/m2
120
Module Output Power (W)
120
Module Output Power (W)
(b) incCond algorithm
160
2
40
100
600W/m2
80
400W/m2
60
40
200W/m2
200W/m2
20
0
20
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
50
0
0
5
10
15
20
25
30
Module Voltage (V)
35
40
45
Figure 46: Traces of MPP tracking on a sunny day (25oC)
On a cloudy day, the irradiance level changes rapidly because of passing clouds.
MPP tracking is supposed to be challenging. Figure 47 shows the trace of PV operating
points for (a) PO algorithm and (b) incCond algorithm. For both algorithms, the deviations
of operating points from the MPPs are obvious when compared to the results of a sunny day.
Between two algorithms, the incCond algorithm is supposed to outperform the PO
algorithm under rapidly changing atmospheric conditions [11]. A close inspection of Figure
47 reveals that the PO algorithm has slightly larger deviations overall and some erratic
behaviors (such as the large deviation pointed by the red arrow). Some erratic traces are,
however, also observable in the plot of the incCond algorithm. In order to make a better
comparison, total electric energy produced during a 12hour period is calculated and
tabulated in Table 43.
64
50
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